Reductive homogeneous spaces of the compact Lie group $G_2$

TL;DR

This paper revisits the classification of reductive homogeneous spaces of the compact Lie group G2, providing geometric models and clarifying relations among these spaces, based on prior algebraic classifications.

Contribution

It offers a geometric perspective on the classification of reductive homogeneous spaces of G2, expanding on previous algebraic results with explicit models and relations.

Findings

Classification of homogeneous reductive spaces of G2 clarified

Geometric models of the spaces constructed

Relations among the spaces explicitly described

Abstract

The first author defended her doctoral thesis Espacios homog\'eneos reductivos y \'algebras no asociativas in 2001, supervised by P. Benito and A. Elduque. This thesis contained the classification of the Lie-Yamaguti algebras with standard enveloping algebra $\mathfrak{g}_2$ over fields of characteristic zero, which in particular gives the classification of the homogeneous reductive spaces of the compact Lie group $G_2$. In this work we revisit this classification from a more geometrical approach. We provide too geometric models of the corresponding homogeneous spaces and make explicit some relations among them.